The probability that a mango is not small is 0.85 (since 15% are small).

The probability that none of the n mangoes is small is given by (0.85)^n.

We need to find the largest n such that (0.85)^n \geq 0.1.

Taking logarithms on both sides, we have:

\(n \leq \frac{\log(0.1)}{\log(0.85)} \approx 14.2\)

\(Since n must be an integer, the largest possible value is n = 14.\)